Termination w.r.t. Q proof of TRS/HMt009.trs

Termination w.r.t. Q of the following Term Rewriting System could not be shown:

Q restricted rewrite system:
The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.

↳ QTRS

  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.

Using Dependency Pairs [1,13] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b3, i3)
START₁(i) -> BUSY₇(F, closed, stop, false, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b2, i2)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b1, i1)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b3, i3)
START₁(i) -> BUSY₇(F, closed, stop, false, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b2, i2)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> OR₂(b1, i1)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [13,14,18] contains 1 SCC with 4 less nodes.

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

BUSY₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
BUSY₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> IDLE₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = max{0, -1}

POL( IDLE₇(x₁, ..., x₇) ) = max{0, 2x₇ - 2}

POL( or₂(x₁, x₂) ) = 2

POL( BF ) = 2

POL( closed ) = 2

POL( F ) = max{0, -1}

POL( empty ) = max{0, -2}

POL( BUSY₇(x₁, ..., x₇) ) = 2x₇

POL( true ) = 2

POL( FS ) = 2

POL( open ) = 2

POL( false ) = 2

POL( stop ) = 1

POL( newbuttons₄(x₁, ..., x₄) ) = x₁ + 2x₂ + x₃ + x₄ + 1

POL( up ) = 1

POL( S ) = 2

POL( B ) = 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

IDLE₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> BUSY₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = max{0, -1}

POL( IDLE₇(x₁, ..., x₇) ) = max{0, 2x₇ - 2}

POL( or₂(x₁, x₂) ) = 0

POL( BF ) = 2

POL( closed ) = 0

POL( F ) = 2

POL( empty ) = max{0, -2}

POL( BUSY₇(x₁, ..., x₇) ) = 2x₇

POL( true ) = 2

POL( FS ) = max{0, -1}

POL( open ) = 2

POL( false ) = max{0, -1}

POL( stop ) = 1

POL( newbuttons₄(x₁, ..., x₄) ) = 2x₃ + x₄ + 2

POL( up ) = 2

POL( S ) = 2

POL( B ) = 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

BUSY₇(S, d, stop, b1, b2, true, i) -> IDLE₇(S, open, stop, b1, b2, false, i)

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = 2

POL( IDLE₇(x₁, ..., x₇) ) = x₆ + 2x₇ + 1

POL( BF ) = 2

POL( closed ) = 2

POL( empty ) = 1

POL( F ) = max{0, -2}

POL( BUSY₇(x₁, ..., x₇) ) = x₆ + 2x₇ + 1

POL( FS ) = 1

POL( true ) = 1

POL( open ) = 2

POL( false ) = max{0, -1}

POL( stop ) = 0

POL( up ) = 2

POL( S ) = max{0, -1}

POL( B ) = 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

BUSY₇(F, d, stop, b1, true, b3, i) -> IDLE₇(F, open, stop, b1, false, b3, i)

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = max{0, -2}

POL( IDLE₇(x₁, ..., x₇) ) = 2x₁ + 2x₂ + 2x₅ + 2

POL( BF ) = 2

POL( closed ) = 0

POL( empty ) = max{0, -1}

POL( F ) = 2

POL( BUSY₇(x₁, ..., x₇) ) = 2x₁ + 2x₂ + 2x₅ + x₇ + 2

POL( FS ) = 2

POL( true ) = 2

POL( open ) = max{0, -2}

POL( false ) = max{0, -2}

POL( stop ) = 2

POL( up ) = 0

POL( S ) = 2

POL( B ) = 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

BUSY₇(B, d, stop, true, b2, b3, i) -> IDLE₇(B, open, stop, false, b2, b3, i)

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = 2

POL( IDLE₇(x₁, ..., x₇) ) = x₂ + x₄ + x₅

POL( BF ) = 1

POL( closed ) = 0

POL( empty ) = 2

POL( F ) = 2

POL( BUSY₇(x₁, ..., x₇) ) = x₄ + x₅

POL( FS ) = 2

POL( true ) = 2

POL( open ) = 1

POL( false ) = max{0, -2}

POL( stop ) = 0

POL( up ) = 2

POL( S ) = 2

POL( B ) = max{0, -2}

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ QDPOrderProof

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [13].

The following pairs can be oriented strictly and are deleted.

BUSY₇(B, open, stop, false, b2, b3, i) -> IDLE₇(B, closed, stop, false, b2, b3, i)
BUSY₇(F, open, stop, b1, false, b3, i) -> IDLE₇(F, closed, stop, b1, false, b3, i)
BUSY₇(S, open, stop, b1, b2, false, i) -> IDLE₇(S, closed, stop, b1, b2, false, i)

The remaining pairs can at least be oriented weakly.

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)

Used ordering: Polynomial Order [17,21] with Interpretation:

POL( down ) = 2

POL( IDLE₇(x₁, ..., x₇) ) = max{0, x₂ + x₄ + 2x₇ - 1}

POL( BF ) = 1

POL( closed ) = 1

POL( empty ) = 1

POL( F ) = 1

POL( BUSY₇(x₁, ..., x₇) ) = max{0, x₂ + x₄ + 2x₇ - 1}

POL( true ) = 1

POL( FS ) = max{0, -1}

POL( open ) = 2

POL( false ) = 1

POL( stop ) = 2

POL( up ) = 2

POL( S ) = 2

POL( B ) = 2

The following usable rules [14] were oriented: none

↳ QTRS

  ↳ DependencyPairsProof

    ↳ QDP

      ↳ DependencyGraphProof

        ↳ QDP

          ↳ QDPOrderProof

            ↳ QDP

              ↳ QDPOrderProof

                ↳ QDP

                  ↳ QDPOrderProof

                    ↳ QDP

                      ↳ QDPOrderProof

                        ↳ QDP

                          ↳ QDPOrderProof

                            ↳ QDP

                              ↳ QDPOrderProof

                                ↳ QDP

Q DP problem:
The TRS P consists of the following rules:

BUSY₇(S, closed, stop, true, false, false, i) -> IDLE₇(S, closed, down, true, false, false, i)
BUSY₇(F, closed, down, b1, false, b3, i) -> IDLE₇(BF, closed, down, b1, false, b3, i)
BUSY₇(B, closed, stop, false, false, true, i) -> IDLE₇(B, closed, up, false, false, true, i)
BUSY₇(FS, closed, down, b1, b2, b3, i) -> IDLE₇(F, closed, down, b1, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, true, i) -> IDLE₇(S, closed, stop, b1, b2, true, i)
BUSY₇(F, closed, up, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
BUSY₇(S, closed, stop, b1, true, false, i) -> IDLE₇(S, closed, down, b1, true, false, i)
BUSY₇(F, closed, up, b1, false, b3, i) -> IDLE₇(FS, closed, up, b1, false, b3, i)
BUSY₇(F, closed, stop, true, false, b3, i) -> IDLE₇(F, closed, down, true, false, b3, i)
BUSY₇(B, closed, stop, false, true, b3, i) -> IDLE₇(B, closed, up, false, true, b3, i)
BUSY₇(F, closed, stop, false, false, true, i) -> IDLE₇(F, closed, up, false, false, true, i)
BUSY₇(BF, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, down, b1, b2, b3, i)
BUSY₇(F, closed, down, b1, true, b3, i) -> IDLE₇(F, closed, stop, b1, true, b3, i)
IDLE₇(fl, d, m, b1, b2, b3, empty) -> BUSY₇(fl, d, m, b1, b2, b3, empty)
BUSY₇(BF, closed, up, b1, b2, b3, i) -> IDLE₇(F, closed, up, b1, b2, b3, i)
BUSY₇(S, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, stop, b1, b2, b3, i)
BUSY₇(B, closed, up, false, b2, b3, i) -> IDLE₇(BF, closed, up, false, b2, b3, i)
BUSY₇(S, closed, down, b1, b2, false, i) -> IDLE₇(FS, closed, down, b1, b2, false, i)
BUSY₇(B, closed, up, true, b2, b3, i) -> IDLE₇(B, closed, stop, true, b2, b3, i)
BUSY₇(FS, closed, up, b1, b2, b3, i) -> IDLE₇(S, closed, up, b1, b2, b3, i)
BUSY₇(B, closed, down, b1, b2, b3, i) -> IDLE₇(B, closed, stop, b1, b2, b3, i)

The TRS R consists of the following rules:

start₁(i) -> busy₇(F, closed, stop, false, false, false, i)
busy₇(BF, d, stop, b1, b2, b3, i) -> incorrect
busy₇(FS, d, stop, b1, b2, b3, i) -> incorrect
busy₇(fl, open, up, b1, b2, b3, i) -> incorrect
busy₇(fl, open, down, b1, b2, b3, i) -> incorrect
busy₇(B, closed, stop, false, false, false, empty) -> correct
busy₇(F, closed, stop, false, false, false, empty) -> correct
busy₇(S, closed, stop, false, false, false, empty) -> correct
busy₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(B, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(F, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i)) -> idle₇(S, closed, stop, false, false, false, newbuttons₄(i1, i2, i3, i))
busy₇(B, open, stop, false, b2, b3, i) -> idle₇(B, closed, stop, false, b2, b3, i)
busy₇(F, open, stop, b1, false, b3, i) -> idle₇(F, closed, stop, b1, false, b3, i)
busy₇(S, open, stop, b1, b2, false, i) -> idle₇(S, closed, stop, b1, b2, false, i)
busy₇(B, d, stop, true, b2, b3, i) -> idle₇(B, open, stop, false, b2, b3, i)
busy₇(F, d, stop, b1, true, b3, i) -> idle₇(F, open, stop, b1, false, b3, i)
busy₇(S, d, stop, b1, b2, true, i) -> idle₇(S, open, stop, b1, b2, false, i)
busy₇(B, closed, down, b1, b2, b3, i) -> idle₇(B, closed, stop, b1, b2, b3, i)
busy₇(S, closed, up, b1, b2, b3, i) -> idle₇(S, closed, stop, b1, b2, b3, i)
busy₇(B, closed, up, true, b2, b3, i) -> idle₇(B, closed, stop, true, b2, b3, i)
busy₇(F, closed, up, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(F, closed, down, b1, true, b3, i) -> idle₇(F, closed, stop, b1, true, b3, i)
busy₇(S, closed, down, b1, b2, true, i) -> idle₇(S, closed, stop, b1, b2, true, i)
busy₇(B, closed, up, false, b2, b3, i) -> idle₇(BF, closed, up, false, b2, b3, i)
busy₇(F, closed, up, b1, false, b3, i) -> idle₇(FS, closed, up, b1, false, b3, i)
busy₇(F, closed, down, b1, false, b3, i) -> idle₇(BF, closed, down, b1, false, b3, i)
busy₇(S, closed, down, b1, b2, false, i) -> idle₇(FS, closed, down, b1, b2, false, i)
busy₇(BF, closed, up, b1, b2, b3, i) -> idle₇(F, closed, up, b1, b2, b3, i)
busy₇(BF, closed, down, b1, b2, b3, i) -> idle₇(B, closed, down, b1, b2, b3, i)
busy₇(FS, closed, up, b1, b2, b3, i) -> idle₇(S, closed, up, b1, b2, b3, i)
busy₇(FS, closed, down, b1, b2, b3, i) -> idle₇(F, closed, down, b1, b2, b3, i)
busy₇(B, closed, stop, false, true, b3, i) -> idle₇(B, closed, up, false, true, b3, i)
busy₇(B, closed, stop, false, false, true, i) -> idle₇(B, closed, up, false, false, true, i)
busy₇(F, closed, stop, true, false, b3, i) -> idle₇(F, closed, down, true, false, b3, i)
busy₇(F, closed, stop, false, false, true, i) -> idle₇(F, closed, up, false, false, true, i)
busy₇(S, closed, stop, b1, true, false, i) -> idle₇(S, closed, down, b1, true, false, i)
busy₇(S, closed, stop, true, false, false, i) -> idle₇(S, closed, down, true, false, false, i)
idle₇(fl, d, m, b1, b2, b3, empty) -> busy₇(fl, d, m, b1, b2, b3, empty)
idle₇(fl, d, m, b1, b2, b3, newbuttons₄(i1, i2, i3, i)) -> busy₇(fl, d, m, or₂(b1, i1), or₂(b2, i2), or₂(b3, i3), i)
or₂(true, b) -> true
or₂(false, b) -> b

Q is empty.
We have to consider all minimal (P,Q,R)-chains.